#187604

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62752	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,11,14,7)(4,15,16,26)(5,10)(6,9)(8,17,23,19)(12,28)(13,25,18,22)(20,24)(21,30)(27,31)(29,32)(33,34)(35,43,46,57)(36,40)(37,42)(38,41)(39,45,53,49)(44,60,62,59)(47,55)(48,54)(50,58)(51,56)(52,61,64,63);(1,3)(2,7)(4,12)(5,18)(6,14)(8,21)(9,11)(10,25)(13,20)(15,28)(16,29)(17,31)(19,30)(22,24)(23,27)(26,32)(33,35)(34,39)(36,44)(37,49)(38,46)(40,52)(41,53)(42,57)(43,56)(45,51)(47,63)(48,62)(50,61)(54,64)(55,59)(58,60);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,23)(10,26)(11,27)(13,31)(14,29)(15,24)(17,20)(18,30)(22,28)(25,32)(33,36,38,48)(34,40,41,54)(35,44,46,62)(37,50,51,47)(39,52,53,64)(42,58,56,55)(43,59,57,60)(45,63,49,61);(2,9)(4,16)(7,11)(10,24)(12,29)(17,19)(22,25)(30,31)(34,41)(39,53)(40,54)(42,56)(43,57)(52,64)(55,58)(59,60)

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,f^b=fhi,g^b=ghi,d^c=dh,e^c=ehi,f^c=fh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94800);